Abstract

Generalizations of Wendroff type integral inequalities with four dependent limits and their discrete analogues are obtained. In applications, these results are used to establish the stability estimates for the solution of the Goursat problem.

1. Introduction

Integral inequalities play a significant role in the theory of ordinary and partial differential equations. They are useful to investigate some properties of the solutions of equations, such as existence, uniqueness, and stability, see for instance [1–6].

Most scientific and technical problems can be solved by using mathematical modelling and new numerical methods. This is based on the mathematical description of real processes and the subsequent solving of the appropriate mathematical problems on the computer. The mathematical models of many scientific and technical problems lead to already known or new problems of partial differential equations. In most of the cases it is difficult to find the exact solutions of the problems for partial differential equations. For this reason discrete methods play a significant role, especially due to the increasing role of mathematical methods of solving problems in various areas of science and engineering. A well-known and widely applied method of approximate solutions for problems of differential equations is the method of difference schemes. Modern computers allow us to implement highly accurate difference schemes. Hence, the task is to construct and investigate highly accurate difference schemes for various types of partial differential equations. The investigation of stability and convergence of these difference schemes is based on the discrete analogues of integral inequalities, see for instance [1, 7–9].

Due to various motivations, several generalizations and applications of Wendroff-type integral inequality have been obtained and used extensively, see for instance [10–12]. In [12], the following generalizations of Wendroff-type integral inequality in two independent variables are obtained.

Theorem 1.1. Assume that and are continuous functions on , and the inequalities
hold, where . Then, for the inequalities:
are satisfied.

Theorem 1.2. Assume that is a continuous function on , and the inequalities:
hold, where is a continuous function on , and increasing with respect to each variable, and are integrable functions on and , respectively. Then, for the inequalities:
are satisfied.

In this paper, generalizations of Wendroff-type integral inequalities in two independent variables with four dependent limits and their discrete analogues are obtained. In applications, these results are used to obtain the stability estimates of solutions for the Goursat problem.

2. Wendroff-type Integral Inequalities with Four Dependent Limits and Their Discrete Analogues

First of all, let us give the discrete analogue of the Gronwall-type integral inequality with two dependent limits. We will need this result in the remaining part of the paper.

Theorem 2.1. Assume that , , , are the sequences of real numbers and the inequalities:
hold. Then, for the inequalities:
are satisfied.

Proof. The proof of (2.2) for follows directly from (2.1). Let us prove (2.2) for . We denote
Then, (2.1) gets the form
Moreover, we have
Then, using (2.3)–(2.5) for , we obtain
So,
Then by induction, we can prove that
for . Since , using (2.4) and the inequality , , we obtain
So, we proved (2.2) for .Let us prove (2.2) for . Using (2.3)–(2.5) for , we have
So,
Then by induction, we can prove that
for . Since , using (2.4) and the inequality , , we obtain
So, we proved (2.2) for .

By putting and passing to limit in the Theorem 2.1, we obtain the following generalization of Gronwall's integral inequality with two dependent limits.

Theorem 2.2. Assume that , are continuous functions on , is an integrable function on , and the inequalities
hold. Then for , the inequalities:
are satisfied.

Now, we consider the generalizations of Wendroff inequality for integrals in two independent variables with four dependent limits and their discrete analogues.

Theorem 2.3. Assume that , , , are sequences of real numbers, and the inequalities:
hold, where , , . Then for , the following inequalities:
are satisfied.

Proof. The proof of (2.17) for , and , follows directly from (2.16). Let us prove (2.17) for , . We denote
Then, (2.16) gets the form:
Furthermore, we have
From (2.18) for , , we have
So,
Then using (2.18)–(2.22) for , , we obtain
So,
Then by induction, we can prove that
Since , , using (2.19) and the inequality , , we obtain
So, we proved (2.17) for , .Using (2.18)–(2.22) for , , we obtain
So,
Then by induction, we can prove that
Since , using (2.19) and the inequality , , we obtain
So, we proved (2.17) for , .

Theorem 2.4. Assume that , , , , , are sequences of real numbers, and the inequalities:
hold, where , . Then for , , , the inequalities:
are satisfied.

Proof. We denote , , . Then, (2.31) takes the form
Next, by denoting
we have
By using Theorem 2.1, we obtain
Inserting (2.35) yields
Now, by denoting
we have
Let us denote the right-hand side of (2.40) by . Then, (2.40) gets the form
Then by using induction, we can prove that
By using Theorem 2.3, we obtain
Finally, by combining (2.39), (2.41), and (2.43), we finish the proof of (2.33).

By putting , and passing to limit as , in Theorems 2.3 and 2.4, we obtain the following two theorems about the generalizations of Wendroff integral inequality with four dependent limits.

Theorem 2.5. Assume that and are continuous functions on , and the inequalities
hold, where . Then for the inequalities
are satisfied.

Theorem 2.6. Assume that is a continuous function on , , and the inequalities:
hold, where is a continuous function on , and increasing with respect to each variable, and are integrable functions on and , respectively. Then for , the inequalities
are satisfied.

Finally, we formulate (without proofs) the generalizations of Wendroff-type inequalities for the integrals in three independent variables with six dependent limits.

Theorem 2.7. Assume that , are continuous functions, and the inequalities:
hold, where . Then for , the inequalities
are satisfied.

Theorem 2.8. Assume that is a continuous function, and the inequalities:
hold, where , , and are integrable functions, is continuous and increasing with respect to each variable. Then for , the inequalities:
are satisfied, where

3. Applications

In applications, we consider the Goursat problem for hyperbolic equations:

A function is called a solution of the Goursat problem (3.1) if the following conditions are satisfied: is twice continuously differentiable on the region , and the derivatives at the endpoints are understood as the appropriate unilateral derivatives;

Theorem 3.1. Assume that the functions and are continuously differentiable and . Let , , and be continuously differentiable functions. Then, there is a unique solution of the problem (3.1) and the stability inequalities:
hold, where and do not depend on and and

The proof of this theorem is based on the formula:
and on the Theorem 2.6.

Now, we consider the difference schemes for approximate solutions of problem (3.1):
where .

Theorem 3.2. For the solution of difference schemes (3.5), the following estimates are satisfied:
where does not depend on , , , , .

The proof of this theorem is based on the following formula:
and on Theorem 2.4.

4. Conclusion

In this paper, generalizations of Wendroff-type inequalities for the integrals in two independent variables with four dependent limits and their discrete analogues are studied. The generalizations of Wendroff-type integral inequalities are used to establish stability estimates for the solution of the Goursat problem. A difference scheme approximately solving the Goursat problem is presented. Bu using the discrete analogues of the generalizations of Wendroff-type integral inequalities, stability estimates for the solution of this difference scheme are established.